# Serre functor of an injective module

Suppose $X=\mathrm{Spec}R$ where $R$ is a noetherian ring. We have the Serre functor $\mathcal{A}(M)$ of a module giving a coherent sheaf. Hartshorne spends a page proving that $\mathcal{A}(I)$ is a flasque sheaf for an injective module $I$. Isn't it true that $\mathcal{A}(I)$ is an injective object in the category of coherent sheaves?

In the end,he uses this to prove that the cohomology of a quasicoherent sheaf vanishes on a noetherian affine scheme. Wouldn't it have been easier simply to assert that $\mathcal{A}(I)$ is injective?

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There is the Abelian category of coherent sheaves, but there is also the larger category of $\mathcal{O}_X$-modules, or even all Abelian sheaves. For some purposes, e.g., for computing $H^1(X,\mathcal{O}_X^\ast)$, it is important to work in the larger category. In any of these categories, flasque sheaves are acyclic for sheaf cohomology. –  Jason Starr Dec 17 '12 at 18:04
He wants to derive the functor of global sections in the category of sheaves of abelian groups, and not in the category of quasi-coherent sheaves. His argument shows that the two notions coincide. Let me give you an example, where they don't: Let $G$ be a $p$-group, let $R$ be the category of all $\mathbb{F}_p$-representations of $G$ and let $V$ be the full subcategory of $R$, consisting of vector spaces with trivial $G$ -action. If you derive the functor $r\mapsto r^G$ (the invariants), then the derived functors in $V$ are zero, and non-zero in $R$. –  labirintas Dec 17 '12 at 18:09
Excellent point! –  Justin Smith Dec 17 '12 at 23:58